Entropy, free energy and entropic forces

Lecture notes, weeks 10–11. Nelson, chapters 6–7

Biological question: if energy is always conserved, how can some devices be more efficient than others? Physical idea: order controls when energy can do useful work, and order is not conserved.

What decides equilibrium?

In mechanics, a system at rest sits where the forces balance, \(\sum_i \vec F_i = -\sum_i\nabla U_i = 0\): it minimizes its potential energy. In thermodynamics, an isolated system goes to the macrostate with the largest number of microstates: it maximizes its entropy \(S = k_B\ln\Omega\).

Most biological systems are not isolated. A protein, a bead or a cell sits in water at fixed temperature \(T\); the water is a huge heat reservoir. If the small system takes energy \(\Delta U\) from the reservoir, the reservoir’s entropy changes by \(-\Delta U/T\). The total entropy change is

\[ \Delta S_\text{tot} = \Delta S - \frac{\Delta U}{T} = -\frac{\Delta(U - TS)}{T} = -\frac{\Delta F}{T}. \]

So maximizing the total entropy is the same as minimizing the Helmholtz free energy \(F = U - TS\) of the small system. At fixed pressure as well (an open test tube, a cell), the work \(P\Delta V\) done on the atmosphere also counts, and the quantity to minimize is the Gibbs free energy \(G = U + PV - TS\).

The two principles are tied together by the thermodynamic identity.

Thermodynamic potentials

Potential	Definition	Identity	Natural variables
Internal energy	\(U\)	\(dU = T\,dS - P\,dV + \mu\, dN\)	\(S, V, N\)
Enthalpy	\(H = U + PV\)	\(dH = T\,dS + V\,dP + \mu\, dN\)	\(S, P, N\)
Helmholtz free energy	\(F = U - TS\)	\(dF = -S\,dT - P\,dV + \mu\, dN\)	\(T, V, N\)
Gibbs free energy	\(G = U + PV - TS\)	\(dG = -S\,dT + V\,dP + \mu\, dN\)	\(T, P, N\)

Each identity gives the other variables as derivatives, for example \(S = -(\partial F/\partial T)_{V,N}\), \(P = -(\partial F/\partial V)_{T,N}\), and

\[ \mu = -T\left(\frac{\partial S}{\partial N}\right)_{U,V} = \left(\frac{\partial F}{\partial N}\right)_{T,V} = \left(\frac{\partial G}{\partial N}\right)_{T,P}. \tag{1}\]

A useful way to read them: \(F\) is the energy needed to create the system, minus what the reservoir supplies as heat; \(G\) also includes the work of making room for it.

What can be measured? \(T\), \(V\), \(P\), \(N\) (by weighing), heat (from a known dissipation, for example \(RI^2\)) and work. \(U\) follows from the first law and \(C_V\) from \(\Delta U/\Delta T\). Entropy is never measured directly: it is an integral, \(S(T_2) - S(T_1) = \int_{T_1}^{T_2} (C_V/T)\, dT\). The chemical potential is a derivative of \(S\), and is found from equilibria, for example osmotic or electrical ones.

Chemical potential

\(\mu\) is the free energy cost of adding one particle. Particles move from high to low \(\mu\), just as heat flows from high to low \(T\). In equilibrium, \(\mu\) of every species that can move between two regions is the same in both.

Ideal gas. At constant \(T\) and \(N\), \(dG = V\,dP = (Nk_BT/P)\,dP\), so \(G - G_0 = Nk_BT\ln(P/P_0)\) and

\[ \mu = \mu_0 + k_BT \ln\frac{P}{P_0} = \mu_0 + k_BT\ln\frac{c}{c_0}. \tag{2}\]

Statistical mechanics gives the same result with the constant filled in: \(F = -Nk_BT\left[\ln\big(V/(N v_Q)\big) + 1\right]\), where \(v_Q\) is the quantum volume of one molecule, so \(\mu = (\partial F/\partial N)_{T,V} = k_BT\ln(c\, v_Q)\).

Ideal mixtures and dilute solutions. Mixing two ideal gases at equal \(T\) and \(P\) increases the volume available to species 1 from \(V_1\) to \(V_1 + V_2\). Then \(\Delta\mu_1 = -k_BT\ln[(V_1+V_2)/V_1] = k_BT\ln x_1\), where \(x_1 = N_1/(N_1+N_2)\) is the mole fraction:

\[ \mu_i(P, T, x_i) = \mu_i^0(P, T) + k_BT\ln x_i . \tag{3}\]

For a dilute solute this is the same as Eq. 2 with \(c\) the solute concentration. A dilute solute behaves like an ideal gas, with the solvent playing the role of empty space. This is the basis of osmosis (see Processes at membranes).

Potential energy in the chemical potential

If each particle also has a potential energy \(U_\text{ext}\) (gravity, a centrifuge, an electric field), adding a particle costs that energy too:

\[ \mu_\text{tot} = \mu_\text{int} + \mu_\text{ext} = \mu_0 + k_BT\ln\frac{c}{c_0} + U_\text{ext}. \]

In equilibrium \(\mu_\text{tot}\) is the same everywhere. For air in gravity, \(U_\text{ext} = mgz\):

\[ \mu_0 + k_BT\ln\frac{P(z)}{P_0} + mgz = \mu_0 \quad\Rightarrow\quad \frac{P(z)}{P_0} = e^{-mgz/k_BT} = e^{-z/z_c},\quad z_c = \frac{k_BT}{mg}. \]

For nitrogen at 288 K, \(z_c \approx 8.7\) km; on top of Mount Everest the pressure is about a third of that at sea level. The same argument with the buoyant mass of a protein gives sedimentation equilibrium (Nelson 5.1, E4), and with \(U_\text{ext} = ze\mathcal V\) for an ion it gives the Nernst potential (Processes at membranes).

The Boltzmann distribution

Put a small system in contact with a large reservoir with total energy \(E_0\). The probability that the small system is in a particular microstate \(i\) with energy \(\varepsilon_i\) is proportional to the number of states the reservoir then has, \(\Omega_R(E_0 - \varepsilon_i) = e^{S_R(E_0-\varepsilon_i)/k_B}\). Since \(\varepsilon_i \ll E_0\), expand \(S_R\) to first order, with \(\partial S_R/\partial E = 1/T\):

\[ P(i) = \frac{1}{Z}\, e^{-\varepsilon_i/k_BT}, \qquad Z = \sum_i e^{-\varepsilon_i/k_BT}. \tag{4}\]

The partition function \(Z\) normalizes the probabilities and contains all the thermodynamics of the small system: \(\langle E\rangle = k_BT^2\, d\ln Z/dT\) and \(F = -k_BT\ln Z\).

Every quadratic term in the energy, like \(\tfrac12 mv_x^2\) or \(\tfrac12 kx^2\), has average \(\tfrac12 k_BT\) (equipartition). This is how you measure the stiffness of something you cannot pull on: watch its thermal fluctuations, \(k = k_BT/\langle\Delta x^2\rangle\).

Two-state systems

Many molecular switches have two states: a channel is open or closed, a protein folded or unfolded, an RNA hairpin zipped or unzipped. If the two states differ by free energy \(\Delta F = \Delta E - T\Delta S\) (each state is itself a group of many microstates, hence the entropy), then

\[ \frac{P_2}{P_1} = e^{-\Delta F/k_BT}, \qquad P_2 = \frac{1}{1 + e^{\Delta F/k_BT}}. \]

Pulling on a molecule. If state 2 is longer by \(\Delta z\) and we pull with a force \(f\), the force does work \(f\Delta z\) when the molecule opens, so \(\Delta F \to \Delta F_0 - f\Delta z\):

\[ P_\text{open}(f) = \frac{1}{1 + e^{(\Delta F_0 - f\Delta z)/k_BT}}. \tag{5}\]

The molecule is open half the time at \(f_{1/2} = \Delta F_0/\Delta z\), and the switch from mostly closed to mostly open happens over a force range of a few \(k_BT/\Delta z\). Optical-tweezer experiments on a single RNA hairpin (Liphardt et al. 2001) find \(\Delta z\approx 18\) nm and \(f_{1/2}\approx 14\) pN. So \(\Delta F_0\approx 60\,k_BT\), and the transition width \(k_BT/\Delta z\approx 0.2\) pN is sharp, as measured. One fit of a curve to Eq. 5 gives both a length and a free energy of a single molecule.

Entropic forces

A force is the derivative of a free energy, \(f = -\partial F/\partial x\), not only of an energy. When the energy does not depend on \(x\), all of the force comes from entropy, \(f = T\,\partial S/\partial x\).

The ideal gas is the simplest example. A gas pushes on a piston although its molecules have no potential energy at all: \(F = -Nk_BT\ln V + \text{const}\), so \(P = -\partial F/\partial V = Nk_BT/V\). The gas pushes because larger volume means more microstates. Osmotic pressure is the same force, exerted by solute particles on a membrane they cannot cross.

A polymer as an entropic spring. Model a chain as \(N\) segments of length \(L\), each pointing left or right (a one-dimensional random walk, Block A). Pulled with force \(f\), each segment is a two-state system with energy \(\mp fL\), so \(\langle x\rangle = NL\tanh(fL/k_BT)\). For small forces \(\langle x\rangle \approx (NL^2/k_BT)\,f\): the chain is a spring with stiffness \(k = k_BT/(NL^2)\). Unlike a metal spring, it gets stiffer when heated, because the restoring force is the thermal drive back towards the many coiled states. Rubber bands behave this way.

The depletion force

Large objects in a suspension of smaller particles (proteins, polymers, micelles) are pushed together. Small spheres of radius \(R\) cannot get closer than \(R\) to a surface (their centres are excluded from a layer of thickness \(R\)). When two flat surfaces are closer than \(2R\), no small particles fit between them. The osmotic pressure \(ck_BT\) of the small particles on the outside is then unbalanced, and pushes the surfaces together.

Equivalently: bringing the surfaces together removes the overlap of their excluded layers and gives the small particles more room. For surfaces of area \(A\) at separation \(d<2R\) the free volume gained is \(A(2R-d)\), so

\[ \Delta F = -c\,k_BT\, A\,(2R - d), \qquad d < 2R . \]

This is a purely entropic attraction, with range set by the size of the small particles and strength by their concentration. In the crowded cytoplasm, with about 30 % of the volume taken by proteins, it is strong enough to help assemble macromolecular complexes and filament bundles (E7).

Free energy transduction

A stretched spring, a concentration difference and an unequal distribution of ions all store free energy. How much useful work can be recovered depends on how the constraint is released. Releasing a big constraint all at once (the spring snaps back against a small load, the membrane ruptures) wastes most of it as heat. Releasing it in many small, controlled steps (a motor that takes one step per ATP, a pump that moves one ion at a time) can come close to the full \(\Delta F\). Biology uses the second strategy. See Nelson section 6.5.3 and E6.